In the previous sections, we proposed a Fourier like basis on directed graphs, as the set of random walk's eigenvectors and determine a frequency analysis by studying the smoothness of the random walk's eigenvectors. We are thus able to construct multiresolution analyses of functions over directed graphs. In a first instance, we propose wavelet frames made of analysis and synthesis graph filter banks. This multi-scale construction is closely related to the diffusion polynomial frames construction [1]} and spectral graph wavelets [2]}. In a second instance, we propose a critically sampled wavelet construction on directed graphs generalizing the diffusion wavelets framework [3]}. In our theoretical constructions, we propose to work in the \(\ell ^2(\mathcal {V})\)  space. We use the linear transformation \(\varphi \)  defined in isometricop to go into the \(\ell ^2(\mathcal {V},\pi )\)  space.
